Frobenius intertwiners for q-difference equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908241924980736 |
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| author | Smirnov, Andrey |
| author_facet | Smirnov, Andrey |
| contents | We consider a class of $q$-hypergeometric equations describing the quantum difference equation for the cotangent bundles over projective spaces $X=T^{*}\mathbb{P}^{n-1}$ . We show that over $\mathbb{Q}_p$ these equations are equipped with the Frobenius action $(q,z)\to (q^p,z^p)$. We obtain an explicit formula for the constant term of the Frobenius intertwiner in terms of the $p$-adic $q$-gamma function of Koblitz. In the limit $q\to 1$ we arrive at the Frobenius structures for the $p$-adic hypergeometric and Bessel differential equations studied by Dwork. In particular, we find closed formulas for $p$-adic constants appearing in works of Dwork and Sperber in terms of $p$-adic zeta functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_00206 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Frobenius intertwiners for q-difference equations Smirnov, Andrey Number Theory High Energy Physics - Theory Mathematical Physics Algebraic Geometry Representation Theory We consider a class of $q$-hypergeometric equations describing the quantum difference equation for the cotangent bundles over projective spaces $X=T^{*}\mathbb{P}^{n-1}$ . We show that over $\mathbb{Q}_p$ these equations are equipped with the Frobenius action $(q,z)\to (q^p,z^p)$. We obtain an explicit formula for the constant term of the Frobenius intertwiner in terms of the $p$-adic $q$-gamma function of Koblitz. In the limit $q\to 1$ we arrive at the Frobenius structures for the $p$-adic hypergeometric and Bessel differential equations studied by Dwork. In particular, we find closed formulas for $p$-adic constants appearing in works of Dwork and Sperber in terms of $p$-adic zeta functions. |
| title | Frobenius intertwiners for q-difference equations |
| topic | Number Theory High Energy Physics - Theory Mathematical Physics Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2406.00206 |